Introduction to Quantum Logic

Chris Heunen

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Overview

I Boolean algebra

I Superposition

I Quantum logic

I Entanglement

I Quantum computation

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Boolean algebra

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Boolean algebra

A Boolean algebra is a set (of “logical propositions”) with

I special elements 0, 1 (“false” and “true”)

I binary operations ∨,∧ (“or” and “and”)

I a unary operation ¬ (“not”)

that satisfy laws:

associativity x ∨ (y ∨ z) = (x ∨ y) ∨ z x ∧ (y ∧ z) = (x ∧ y) ∧ zcommutativity x ∨ y = y ∨ x x ∧ y = y ∧ xidentity x ∨ 0 = x x ∧ 1 = xannihilation x ∨ 1 = 1 x ∧ 0 = 0idempotence x ∨ x = x x ∧ x = xabsorption x ∧ (x ∨ y) = x x ∨ (x ∧ y) = xcomplementation x ∧ ¬x = 0 x ∨ ¬x = 1de Morgan ¬(x ∨ y) = ¬x ∧ ¬y ¬(x ∧ y) = ¬x ∨ ¬ydouble negation ¬(¬x) = xdistributivity x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

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Boolean algebra

A Boolean algebra is a set (of “logical propositions”) with

I special elements 0, 1 (“false” and “true”)

I binary operations ∨,∧ (“or” and “and”)

I a unary operation ¬ (“not”)

that satisfy laws:

associativity x ∨ (y ∨ z) = (x ∨ y) ∨ z x ∧ (y ∧ z) = (x ∧ y) ∧ zcommutativity x ∨ y = y ∨ x x ∧ y = y ∧ xidentity x ∨ 0 = x x ∧ 1 = xannihilation x ∨ 1 = 1 x ∧ 0 = 0idempotence x ∨ x = x x ∧ x = xabsorption x ∧ (x ∨ y) = x x ∨ (x ∧ y) = xcomplementation x ∧ ¬x = 0 x ∨ ¬x = 1de Morgan ¬(x ∨ y) = ¬x ∧ ¬y ¬(x ∧ y) = ¬x ∨ ¬ydouble negation ¬(¬x) = xdistributivity x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

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Venn diagram

A

B

C

“logical proposition” ∼ subset“and” ∼ intersection

“or” ∼ union“not” ∼ complement

“true” ∼ whole set“false” ∼ empty set

Think of a “logical proposition” as the set of states in which it is trueThe larger the subset, the “more true” the proposition

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Venn diagram

A

B

C

“logical proposition” ∼ subset“and” ∼ intersection

“or” ∼ union“not” ∼ complement

“true” ∼ whole set“false” ∼ empty set

Think of a “logical proposition” as the set of states in which it is trueThe larger the subset, the “more true” the proposition

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Hasse diagramEvery Boolean algebra has a partial order

x ≤ y ⇐⇒ x ∧ y = x

with greatest lower bounds, least upper bounds, and complements.Conversely, every such partial order gives a Boolean algebra.

{}

{, ,

}

{ } { } { }

{,

} {,

} {,}

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Hasse diagramEvery Boolean algebra has a partial order

x ≤ y ⇐⇒ x ∧ y = x

with greatest lower bounds, least upper bounds, and complements.

Conversely, every such partial order gives a Boolean algebra.

{}

{, ,

}

{ } { } { }

{,

} {,

} {,}

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Hasse diagramEvery Boolean algebra has a partial order

x ≤ y ⇐⇒ x ∧ y = x

with greatest lower bounds, least upper bounds, and complements.Conversely, every such partial order gives a Boolean algebra.

{}

{, ,

}

{ } { } { }

{,

} {,

} {,}

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Hasse diagramEvery Boolean algebra has a partial order

x ≤ y ⇐⇒ x ∧ y = x

with greatest lower bounds, least upper bounds, and complements.Conversely, every such partial order gives a Boolean algebra.

{}

{, ,

}

{ } { } { }

{,

} {,

} {,}

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Implication

Can axiomatise Boolean algebra in terms of ∧,∨, or in terms of ≤,or in terms of →:

x ∧ y ≤ z ⇐⇒ x ≤ y → z

where (y → z) = (¬y ∨ z)

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Quantum information

I Boolean logic governs propositions and states.

I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)

I Quantum information is weird:I superpositionI entanglement

I Quantum computers use this weirdness in a positive wayto achieve more than classical computers

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Quantum information

I Boolean logic governs propositions and states.

I Computers manipulate information(information stored on physical system)

Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)

I Quantum information is weird:I superpositionI entanglement

I Quantum computers use this weirdness in a positive wayto achieve more than classical computers

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Quantum information

I Boolean logic governs propositions and states.

I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)

I Quantum information is weird:I superpositionI entanglement

I Quantum computers use this weirdness in a positive wayto achieve more than classical computers

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Quantum information

I Boolean logic governs propositions and states.

I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)

I Quantum information is weird:I superpositionI entanglement

I Quantum computers use this weirdness in a positive wayto achieve more than classical computers

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Quantum information

I Boolean logic governs propositions and states.

I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)

I Quantum information is weird:I superpositionI entanglement

I Quantum computers use this weirdness in a positive wayto achieve more than classical computers

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States and propositions

Physical system has set of statesProposition about physical system is subset

Quantum system has space of statesProposition about quantum system is subspace

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States and propositions

Physical system has set of statesProposition about physical system is subset

Quantum system has space of statesProposition about quantum system is subspace

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Quantum weirdness: superposition

Classical bits

(what went in comes out)

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Superposition

Quantum bits

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Superposition

Quantum bits

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Superposition

Quantum bits

(if you open different door than you closed, random colour comesout)

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Qubits

Quantum bit has state space R2

Could be

(10

), could be

(01

), or could be “in between”

(ab

).

You can ask for the value of a quantum bit in many ways,

using any angle θ. Say

(cos θ − sin θsin θ cos θ

)(ab

)=

(cd

).

Get answer 0 with probability c2, answer 1 with probability d2.

So propositions are the subspaces

{(t cos θt sin θ

): t ∈ R

},

{(00

)}, R2.

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Qubits

Quantum bit has state space R2

Could be

(10

), could be

(01

), or could be “in between”

(ab

).

You can ask for the value of a quantum bit in many ways,

using any angle θ. Say

(cos θ − sin θsin θ cos θ

)(ab

)=

(cd

).

Get answer 0 with probability c2, answer 1 with probability d2.

So propositions are the subspaces

{(t cos θt sin θ

): t ∈ R

},

{(00

)}, R2.

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Qubits

Quantum bit has state space R2

Could be

(10

), could be

(01

), or could be “in between”

(ab

).

You can ask for the value of a quantum bit in many ways,

using any angle θ. Say

(cos θ − sin θsin θ cos θ

)(ab

)=

(cd

).

Get answer 0 with probability c2, answer 1 with probability d2.

So propositions are the subspaces

{(t cos θt sin θ

): t ∈ R

},

{(00

)}, R2.

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Distributivity

(or

)and

6=(and

)or(

and)

tea

coffeebiscuit

nothing

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Distributivity

(or

)and

6=(and

)or(

and)

tea

coffeebiscuit

nothing

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Orthomodularity

I There is still order: ≤ is set inclusion.

I There are still least upper bounds, greatest lower bounds.

I There is still negation: ¬R2 =

{(00

)}, and

¬{(

t cos θt sin θ

): t ∈ R

}=

{(t cos(θ + π/2)t sin(θ + π/2)

): t ∈ R

}I The orthomodular law still holds:

x ≤ y =⇒ x ∨ (¬x ∧ y) = y

(distributivity x ∨ (z ∧ y) = (x ∨ z) ∧ (x ∨ y) for z = ¬x, x ≤ y)

I Quantum logic is study of partial orders with 0,1, least upperbounds, greatest lower bounds, complements, satisfyingorthomodular law.

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Implication

There is no good notion of quantum implication. Best we can do is

x& y ≤ z ⇐⇒ x ≤ y → z

where (x& y) = (x ∨ ¬y) ∧ yand (y → z) = ¬y ∨ (y ∧ z).

Here (x& y) = (x ∧ y) when x ≤ ¬y.

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Quantum computation

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Entanglement

2 quantum bits

random random

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Entanglement

2 quantum bits

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Entanglement

2 quantum bits

(same door, same colour!)

I information stored entirely in correlations, not locally!

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Entanglement

classical correlations

quantum correlations

I But: only one way to look at socks,but two ways to look in box!

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Entanglement

classical correlations quantum correlations

I But: only one way to look at socks,but two ways to look in box!

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Entanglement

classical correlations quantum correlations

I But: only one way to look at socks,but two ways to look in box!

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Tensor products

I State space of n bits is product of state spaces of individual bits.

I Product of n qubits R2 × · · · × R2 ' R2n has dimension 2n.

I Instead, use tensor product R2 ⊗ · · · ⊗ R2, with dimension 2n.

I Has many entangled states not in the product.

(ab

)×(cd

)=

abcd

but

(ab

)⊗(cd

)=

acadbcbd

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Quantum computation speed-up

10 classical bits: only 210 = 1024 possibilities

need 10 numbers to describe one possibility:

(all independent)

10 quantum bits:

need ∼ 1000 numbers to describe a single possibility!(many correlations)

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Quantum computation speed-up

10 classical bits: only 210 = 1024 possibilities

need 10 numbers to describe one possibility:

(all independent)

10 quantum bits:

need ∼ 1000 numbers to describe a single possibility!(many correlations)

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Quantum computation speed-up

10 classical bits: only 210 = 1024 possibilities

need 10 numbers to describe one possibility:

(all independent)

10 quantum bits:

need ∼ 1000 numbers to describe a single possibility!(many correlations)

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Deutsch–Josza

I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,

or f always gives the same output.Question: find out which.

I Classical algorithm requires n+ 1 calls to f .

I Quantum algorithm can do it in 1 step!(Caveat: “oracle” f needs to be quantum to start with)

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Deutsch–Josza

I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,

or f always gives the same output.Question: find out which.

I Classical algorithm requires n+ 1 calls to f .

I Quantum algorithm can do it in 1 step!(Caveat: “oracle” f needs to be quantum to start with)

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Deutsch–Josza

I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,

or f always gives the same output.Question: find out which.

I Classical algorithm requires n+ 1 calls to f .

I Quantum algorithm can do it in 1 step!

(Caveat: “oracle” f needs to be quantum to start with)

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Deutsch–Josza

I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,

or f always gives the same output.Question: find out which.

I Classical algorithm requires n+ 1 calls to f .

I Quantum algorithm can do it in 1 step!(Caveat: “oracle” f needs to be quantum to start with)

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Deutsch–Josza

I Start with

(10

)⊗ · · · ⊗

(10

)⊗(

01

).

I Apply H ⊗ · · · ⊗H ⊗H, where H =

(cosπ/2 − sinπ/2sinπ/2 cosπ/2

)I Apply f

I Apply H ⊗ · · · ⊗HI Measure with angle 0

I Answer 1 with certainty if f was constant, 0 if f was balanced

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Deutsch–Josza

Correctness proof in vector space notation:

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Summary

I Quantum weirdness: superposition, entanglement

I Quantum computation can use weirdness

I Quantum logic has to deal with weirdness

Take-home message:

I Information is physical

I Logic is physical

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